What is the minimal and the characteristic polynomial of T(A) = A^t - A?

Question

We have the following trasformation: $$T:M_n(\mathbb C) \to M_n(\mathbb C) \\ T(A) = A^t-A $$ Let $E$ be the standard basis of $M_n(\mathbb C)$. What is the minimal and the characteristic polynomial of $T$? I tried to solve for $M_2(\mathbb C)$, so the matrix of $T$ is:$$ \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & -1 & 1 & 0 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ \end{pmatrix} $$ I'm not sure if this is right and I'll be greatful if you help me to generalize this for $M_n(\mathbb C)$.

Answer 1

Every matrix can be decomposed into its symmetric and antisymmetric part: $$A = A_+ + A_-\quad\text{ where }\quad A_{\pm} = \frac12(A\pm A^T)$$ Notice $T(A_+) = 0$ and $T(A_-) = -2A_-$ and the dimensions for symmetric and anti-symmetric matrices are $\frac{n(n+1)}{2}$ and $\frac{n(n-1)}{2}$. The matrix for $T$ can be diagonalised to one with $\frac{n(n+1)}{2}$ entries of $0$ and $\frac{n(n-1)}{2}$ entries of $-2$ on its diagonal. This means

• the minimal polynomial for $T$ is $\lambda(\lambda+2)$

• the characteristic polynomial for $T$ is $\lambda^{\frac{n(n+1)}{2}}(\lambda+2)^{\frac{n(n-1)}{2}}$.

Answer 2

$T^2(A)=T(A^t-A)=(A^t-A)^t-(A^t-A)$ $=A-A^t-A^t+A=-2T(A)$. $X^2+2X$ is the minimal polynomial.